1,367 research outputs found
Formal Probabilistic Analysis of a Wireless Sensor Network for Forest Fire Detection
Wireless Sensor Networks (WSNs) have been widely explored for forest fire
detection, which is considered a fatal threat throughout the world. Energy
conservation of sensor nodes is one of the biggest challenges in this context
and random scheduling is frequently applied to overcome that. The performance
analysis of these random scheduling approaches is traditionally done by
paper-and-pencil proof methods or simulation. These traditional techniques
cannot ascertain 100% accuracy, and thus are not suitable for analyzing a
safety-critical application like forest fire detection using WSNs. In this
paper, we propose to overcome this limitation by applying formal probabilistic
analysis using theorem proving to verify scheduling performance of a real-world
WSN for forest fire detection using a k-set randomized algorithm as an energy
saving mechanism. In particular, we formally verify the expected values of
coverage intensity, the upper bound on the total number of disjoint subsets,
for a given coverage intensity, and the lower bound on the total number of
nodes.Comment: In Proceedings SCSS 2012, arXiv:1307.802
Towards the Formal Reliability Analysis of Oil and Gas Pipelines
It is customary to assess the reliability of underground oil and gas
pipelines in the presence of excessive loading and corrosion effects to ensure
a leak-free transport of hazardous materials. The main idea behind this
reliability analysis is to model the given pipeline system as a Reliability
Block Diagram (RBD) of segments such that the reliability of an individual
pipeline segment can be represented by a random variable. Traditionally,
computer simulation is used to perform this reliability analysis but it
provides approximate results and requires an enormous amount of CPU time for
attaining reasonable estimates. Due to its approximate nature, simulation is
not very suitable for analyzing safety-critical systems like oil and gas
pipelines, where even minor analysis flaws may result in catastrophic
consequences. As an accurate alternative, we propose to use a
higher-order-logic theorem prover (HOL) for the reliability analysis of
pipelines. As a first step towards this idea, this paper provides a
higher-order-logic formalization of reliability and the series RBD using the
HOL theorem prover. For illustration, we present the formal analysis of a
simple pipeline that can be modeled as a series RBD of segments with
exponentially distributed failure times.Comment: 15 page
Automatic Probabilistic Program Verification through Random Variable Abstraction
The weakest pre-expectation calculus has been proved to be a mature theory to
analyze quantitative properties of probabilistic and nondeterministic programs.
We present an automatic method for proving quantitative linear properties on
any denumerable state space using iterative backwards fixed point calculation
in the general framework of abstract interpretation. In order to accomplish
this task we present the technique of random variable abstraction (RVA) and we
also postulate a sufficient condition to achieve exact fixed point computation
in the abstract domain. The feasibility of our approach is shown with two
examples, one obtaining the expected running time of a probabilistic program,
and the other the expected gain of a gambling strategy.
Our method works on general guarded probabilistic and nondeterministic
transition systems instead of plain pGCL programs, allowing us to easily model
a wide range of systems including distributed ones and unstructured programs.
We present the operational and weakest precondition semantics for this programs
and prove its equivalence
Formal verification of higher-order probabilistic programs
Probabilistic programming provides a convenient lingua franca for writing
succinct and rigorous descriptions of probabilistic models and inference tasks.
Several probabilistic programming languages, including Anglican, Church or
Hakaru, derive their expressiveness from a powerful combination of continuous
distributions, conditioning, and higher-order functions. Although very
important for practical applications, these combined features raise fundamental
challenges for program semantics and verification. Several recent works offer
promising answers to these challenges, but their primary focus is on semantical
issues.
In this paper, we take a step further and we develop a set of program logics,
named PPV, for proving properties of programs written in an expressive
probabilistic higher-order language with continuous distributions and operators
for conditioning distributions by real-valued functions. Pleasingly, our
program logics retain the comfortable reasoning style of informal proofs thanks
to carefully selected axiomatizations of key results from probability theory.
The versatility of our logics is illustrated through the formal verification of
several intricate examples from statistics, probabilistic inference, and
machine learning. We further show the expressiveness of our logics by giving
sound embeddings of existing logics. In particular, we do this in a parametric
way by showing how the semantics idea of (unary and relational) TT-lifting can
be internalized in our logics. The soundness of PPV follows by interpreting
programs and assertions in quasi-Borel spaces (QBS), a recently proposed
variant of Borel spaces with a good structure for interpreting higher order
probabilistic programs
Using Theorem Proving to Verify Expectation and Variance for Discrete Random Variables
Statistical quantities, such as expectation (mean) and variance, play a vital role in the present age probabilistic analysis. In this paper, we present some formalization of expectation theory that can be used to verify the expectation and variance characteristics of discrete random variables within the HOL theorem prover. The motivation behind this is the ability to perform error free probabilistic analysis, which in turn can be very useful for the performance and reliability analysis of systems used in safety-critical domains, such as space travel, medicine and military. We first present a formal definition of expectation of a function of a discrete random variable. Building upon this definition, we formalize the mathematical concept of variance and verify some classical properties of expectation and variance in HOL. We then utilize these formal definitions to verify the expectation and variance characteristics of the Geometric random variable. In order to demonstrate the practical effectiveness of the formalization presented in this paper, we also present the probabilistic analysis of the Coupon Collector’s problem in HOL
Formal verification of tail distribution bounds in the HOL theorem prover
Tail distribution bounds play a major role in the estimation of failure probabilities in performance and reliability analysis of systems. They are usually estimated using Markov's and Chebyshev's inequalities, which represent tail distribution bounds for a random variable in terms of its mean or variance. This paper presents the formal verification of Markov's and Chebyshev's inequalities for discrete random variables using a higher-order-logic theorem prover. The paper also provides the formal verification of mean and variance relations for some of the widely used discrete random variables, such as Uniform(m), Bernoulli(p), Geometric(p) and Binomial(m, p) random variables. This infrastructure allows us to precisely reason about the tail distribution properties and thus turns out to be quite useful for the analysis of systems used in safety-critical domains, such as space, medicine or transportation. For illustration purposes, we present the performance analysis of the coupon collector's problem, a well-known commercially used algorithm
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